Properties

Label 13.4.a.b.1.1
Level $13$
Weight $4$
Character 13.1
Self dual yes
Analytic conductor $0.767$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.767024830075\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 13.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{2} +8.68466 q^{3} -5.56155 q^{4} -3.56155 q^{5} -13.5616 q^{6} -27.1771 q^{7} +21.1771 q^{8} +48.4233 q^{9} +O(q^{10})\) \(q-1.56155 q^{2} +8.68466 q^{3} -5.56155 q^{4} -3.56155 q^{5} -13.5616 q^{6} -27.1771 q^{7} +21.1771 q^{8} +48.4233 q^{9} +5.56155 q^{10} +15.2614 q^{11} -48.3002 q^{12} -13.0000 q^{13} +42.4384 q^{14} -30.9309 q^{15} +11.4233 q^{16} +44.5464 q^{17} -75.6155 q^{18} +23.9697 q^{19} +19.8078 q^{20} -236.024 q^{21} -23.8314 q^{22} +122.739 q^{23} +183.916 q^{24} -112.315 q^{25} +20.3002 q^{26} +186.054 q^{27} +151.147 q^{28} -219.909 q^{29} +48.3002 q^{30} +27.0928 q^{31} -187.255 q^{32} +132.540 q^{33} -69.5616 q^{34} +96.7926 q^{35} -269.309 q^{36} +94.1922 q^{37} -37.4299 q^{38} -112.901 q^{39} -75.4233 q^{40} -160.354 q^{41} +368.563 q^{42} -151.302 q^{43} -84.8769 q^{44} -172.462 q^{45} -191.663 q^{46} +466.948 q^{47} +99.2074 q^{48} +395.594 q^{49} +175.386 q^{50} +386.870 q^{51} +72.3002 q^{52} -120.847 q^{53} -290.533 q^{54} -54.3542 q^{55} -575.531 q^{56} +208.169 q^{57} +343.400 q^{58} -439.633 q^{59} +172.024 q^{60} -137.305 q^{61} -42.3068 q^{62} -1316.00 q^{63} +201.022 q^{64} +46.3002 q^{65} -206.968 q^{66} +512.280 q^{67} -247.747 q^{68} +1065.94 q^{69} -151.147 q^{70} +410.719 q^{71} +1025.46 q^{72} -308.004 q^{73} -147.086 q^{74} -975.420 q^{75} -133.309 q^{76} -414.759 q^{77} +176.300 q^{78} -586.462 q^{79} -40.6847 q^{80} +308.386 q^{81} +250.401 q^{82} +1354.20 q^{83} +1312.66 q^{84} -158.654 q^{85} +236.266 q^{86} -1909.84 q^{87} +323.191 q^{88} +439.882 q^{89} +269.309 q^{90} +353.302 q^{91} -682.617 q^{92} +235.292 q^{93} -729.164 q^{94} -85.3693 q^{95} -1626.24 q^{96} -1511.27 q^{97} -617.740 q^{98} +739.006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{3} - 7 q^{4} - 3 q^{5} - 23 q^{6} - 9 q^{7} - 3 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{3} - 7 q^{4} - 3 q^{5} - 23 q^{6} - 9 q^{7} - 3 q^{8} + 35 q^{9} + 7 q^{10} + 80 q^{11} - 43 q^{12} - 26 q^{13} + 89 q^{14} - 33 q^{15} - 39 q^{16} + 19 q^{17} - 110 q^{18} - 84 q^{19} + 19 q^{20} - 303 q^{21} + 142 q^{22} + 196 q^{23} + 273 q^{24} - 237 q^{25} - 13 q^{26} + 335 q^{27} + 125 q^{28} - 44 q^{29} + 43 q^{30} - 86 q^{31} - 123 q^{32} - 106 q^{33} - 135 q^{34} + 107 q^{35} - 250 q^{36} + 209 q^{37} - 314 q^{38} - 65 q^{39} - 89 q^{40} - 230 q^{41} + 197 q^{42} + 287 q^{43} - 178 q^{44} - 180 q^{45} - 4 q^{46} + 435 q^{47} + 285 q^{48} + 383 q^{49} - 144 q^{50} + 481 q^{51} + 91 q^{52} - 118 q^{53} + 91 q^{54} - 18 q^{55} - 1015 q^{56} + 606 q^{57} + 794 q^{58} - 368 q^{59} + 175 q^{60} - 1058 q^{61} - 332 q^{62} - 1560 q^{63} + 769 q^{64} + 39 q^{65} - 818 q^{66} + 68 q^{67} - 211 q^{68} + 796 q^{69} - 125 q^{70} - 131 q^{71} + 1350 q^{72} + 456 q^{73} + 147 q^{74} - 516 q^{75} + 22 q^{76} + 762 q^{77} + 299 q^{78} - 1008 q^{79} - 69 q^{80} + 122 q^{81} + 72 q^{82} + 1958 q^{83} + 1409 q^{84} - 173 q^{85} + 1359 q^{86} - 2558 q^{87} - 1242 q^{88} - 720 q^{89} + 250 q^{90} + 117 q^{91} - 788 q^{92} + 652 q^{93} - 811 q^{94} - 146 q^{95} - 1863 q^{96} - 928 q^{97} - 650 q^{98} - 130 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −0.552092 −0.276046 0.961144i \(-0.589024\pi\)
−0.276046 + 0.961144i \(0.589024\pi\)
\(3\) 8.68466 1.67136 0.835682 0.549214i \(-0.185073\pi\)
0.835682 + 0.549214i \(0.185073\pi\)
\(4\) −5.56155 −0.695194
\(5\) −3.56155 −0.318555 −0.159277 0.987234i \(-0.550916\pi\)
−0.159277 + 0.987234i \(0.550916\pi\)
\(6\) −13.5616 −0.922747
\(7\) −27.1771 −1.46742 −0.733712 0.679460i \(-0.762214\pi\)
−0.733712 + 0.679460i \(0.762214\pi\)
\(8\) 21.1771 0.935904
\(9\) 48.4233 1.79346
\(10\) 5.56155 0.175872
\(11\) 15.2614 0.418316 0.209158 0.977882i \(-0.432928\pi\)
0.209158 + 0.977882i \(0.432928\pi\)
\(12\) −48.3002 −1.16192
\(13\) −13.0000 −0.277350
\(14\) 42.4384 0.810154
\(15\) −30.9309 −0.532421
\(16\) 11.4233 0.178489
\(17\) 44.5464 0.635535 0.317767 0.948169i \(-0.397067\pi\)
0.317767 + 0.948169i \(0.397067\pi\)
\(18\) −75.6155 −0.990153
\(19\) 23.9697 0.289422 0.144711 0.989474i \(-0.453775\pi\)
0.144711 + 0.989474i \(0.453775\pi\)
\(20\) 19.8078 0.221458
\(21\) −236.024 −2.45260
\(22\) −23.8314 −0.230949
\(23\) 122.739 1.11273 0.556365 0.830938i \(-0.312196\pi\)
0.556365 + 0.830938i \(0.312196\pi\)
\(24\) 183.916 1.56423
\(25\) −112.315 −0.898523
\(26\) 20.3002 0.153123
\(27\) 186.054 1.32615
\(28\) 151.147 1.02014
\(29\) −219.909 −1.40814 −0.704071 0.710130i \(-0.748636\pi\)
−0.704071 + 0.710130i \(0.748636\pi\)
\(30\) 48.3002 0.293946
\(31\) 27.0928 0.156968 0.0784840 0.996915i \(-0.474992\pi\)
0.0784840 + 0.996915i \(0.474992\pi\)
\(32\) −187.255 −1.03445
\(33\) 132.540 0.699158
\(34\) −69.5616 −0.350874
\(35\) 96.7926 0.467455
\(36\) −269.309 −1.24680
\(37\) 94.1922 0.418516 0.209258 0.977860i \(-0.432895\pi\)
0.209258 + 0.977860i \(0.432895\pi\)
\(38\) −37.4299 −0.159788
\(39\) −112.901 −0.463553
\(40\) −75.4233 −0.298137
\(41\) −160.354 −0.610808 −0.305404 0.952223i \(-0.598791\pi\)
−0.305404 + 0.952223i \(0.598791\pi\)
\(42\) 368.563 1.35406
\(43\) −151.302 −0.536589 −0.268295 0.963337i \(-0.586460\pi\)
−0.268295 + 0.963337i \(0.586460\pi\)
\(44\) −84.8769 −0.290811
\(45\) −172.462 −0.571314
\(46\) −191.663 −0.614329
\(47\) 466.948 1.44918 0.724589 0.689181i \(-0.242030\pi\)
0.724589 + 0.689181i \(0.242030\pi\)
\(48\) 99.2074 0.298320
\(49\) 395.594 1.15333
\(50\) 175.386 0.496067
\(51\) 386.870 1.06221
\(52\) 72.3002 0.192812
\(53\) −120.847 −0.313199 −0.156600 0.987662i \(-0.550053\pi\)
−0.156600 + 0.987662i \(0.550053\pi\)
\(54\) −290.533 −0.732158
\(55\) −54.3542 −0.133257
\(56\) −575.531 −1.37337
\(57\) 208.169 0.483730
\(58\) 343.400 0.777424
\(59\) −439.633 −0.970090 −0.485045 0.874489i \(-0.661197\pi\)
−0.485045 + 0.874489i \(0.661197\pi\)
\(60\) 172.024 0.370136
\(61\) −137.305 −0.288198 −0.144099 0.989563i \(-0.546028\pi\)
−0.144099 + 0.989563i \(0.546028\pi\)
\(62\) −42.3068 −0.0866609
\(63\) −1316.00 −2.63176
\(64\) 201.022 0.392621
\(65\) 46.3002 0.0883513
\(66\) −206.968 −0.386000
\(67\) 512.280 0.934104 0.467052 0.884230i \(-0.345316\pi\)
0.467052 + 0.884230i \(0.345316\pi\)
\(68\) −247.747 −0.441820
\(69\) 1065.94 1.85977
\(70\) −151.147 −0.258078
\(71\) 410.719 0.686526 0.343263 0.939239i \(-0.388468\pi\)
0.343263 + 0.939239i \(0.388468\pi\)
\(72\) 1025.46 1.67850
\(73\) −308.004 −0.493823 −0.246912 0.969038i \(-0.579416\pi\)
−0.246912 + 0.969038i \(0.579416\pi\)
\(74\) −147.086 −0.231060
\(75\) −975.420 −1.50176
\(76\) −133.309 −0.201205
\(77\) −414.759 −0.613847
\(78\) 176.300 0.255924
\(79\) −586.462 −0.835217 −0.417608 0.908627i \(-0.637132\pi\)
−0.417608 + 0.908627i \(0.637132\pi\)
\(80\) −40.6847 −0.0568585
\(81\) 308.386 0.423027
\(82\) 250.401 0.337222
\(83\) 1354.20 1.79088 0.895440 0.445182i \(-0.146861\pi\)
0.895440 + 0.445182i \(0.146861\pi\)
\(84\) 1312.66 1.70503
\(85\) −158.654 −0.202453
\(86\) 236.266 0.296247
\(87\) −1909.84 −2.35352
\(88\) 323.191 0.391503
\(89\) 439.882 0.523904 0.261952 0.965081i \(-0.415634\pi\)
0.261952 + 0.965081i \(0.415634\pi\)
\(90\) 269.309 0.315418
\(91\) 353.302 0.406990
\(92\) −682.617 −0.773563
\(93\) 235.292 0.262351
\(94\) −729.164 −0.800080
\(95\) −85.3693 −0.0921969
\(96\) −1626.24 −1.72894
\(97\) −1511.27 −1.58192 −0.790959 0.611869i \(-0.790418\pi\)
−0.790959 + 0.611869i \(0.790418\pi\)
\(98\) −617.740 −0.636747
\(99\) 739.006 0.750231
\(100\) 624.648 0.624648
\(101\) 336.260 0.331278 0.165639 0.986186i \(-0.447031\pi\)
0.165639 + 0.986186i \(0.447031\pi\)
\(102\) −604.118 −0.586438
\(103\) 322.712 0.308716 0.154358 0.988015i \(-0.450669\pi\)
0.154358 + 0.988015i \(0.450669\pi\)
\(104\) −275.302 −0.259573
\(105\) 840.611 0.781288
\(106\) 188.708 0.172915
\(107\) 1434.62 1.29617 0.648083 0.761570i \(-0.275571\pi\)
0.648083 + 0.761570i \(0.275571\pi\)
\(108\) −1034.75 −0.921933
\(109\) 849.147 0.746179 0.373089 0.927795i \(-0.378298\pi\)
0.373089 + 0.927795i \(0.378298\pi\)
\(110\) 84.8769 0.0735699
\(111\) 818.027 0.699493
\(112\) −310.452 −0.261919
\(113\) 1614.53 1.34409 0.672044 0.740511i \(-0.265417\pi\)
0.672044 + 0.740511i \(0.265417\pi\)
\(114\) −325.066 −0.267064
\(115\) −437.140 −0.354465
\(116\) 1223.04 0.978931
\(117\) −629.503 −0.497415
\(118\) 686.509 0.535579
\(119\) −1210.64 −0.932599
\(120\) −655.026 −0.498295
\(121\) −1098.09 −0.825012
\(122\) 214.409 0.159112
\(123\) −1392.62 −1.02088
\(124\) −150.678 −0.109123
\(125\) 845.211 0.604784
\(126\) 2055.01 1.45297
\(127\) 865.174 0.604502 0.302251 0.953228i \(-0.402262\pi\)
0.302251 + 0.953228i \(0.402262\pi\)
\(128\) 1184.13 0.817683
\(129\) −1314.01 −0.896836
\(130\) −72.3002 −0.0487780
\(131\) −281.400 −0.187680 −0.0938400 0.995587i \(-0.529914\pi\)
−0.0938400 + 0.995587i \(0.529914\pi\)
\(132\) −737.127 −0.486050
\(133\) −651.426 −0.424705
\(134\) −799.953 −0.515712
\(135\) −662.641 −0.422452
\(136\) 943.363 0.594799
\(137\) −2641.43 −1.64725 −0.823624 0.567137i \(-0.808051\pi\)
−0.823624 + 0.567137i \(0.808051\pi\)
\(138\) −1664.53 −1.02677
\(139\) −1998.64 −1.21958 −0.609791 0.792562i \(-0.708747\pi\)
−0.609791 + 0.792562i \(0.708747\pi\)
\(140\) −538.317 −0.324972
\(141\) 4055.28 2.42210
\(142\) −641.359 −0.379026
\(143\) −198.398 −0.116020
\(144\) 553.153 0.320112
\(145\) 783.218 0.448570
\(146\) 480.964 0.272636
\(147\) 3435.60 1.92764
\(148\) −523.855 −0.290950
\(149\) −1752.98 −0.963824 −0.481912 0.876220i \(-0.660058\pi\)
−0.481912 + 0.876220i \(0.660058\pi\)
\(150\) 1523.17 0.829109
\(151\) −2794.64 −1.50613 −0.753063 0.657949i \(-0.771424\pi\)
−0.753063 + 0.657949i \(0.771424\pi\)
\(152\) 507.608 0.270871
\(153\) 2157.08 1.13980
\(154\) 647.669 0.338900
\(155\) −96.4924 −0.0500030
\(156\) 627.902 0.322259
\(157\) 3244.87 1.64949 0.824743 0.565508i \(-0.191320\pi\)
0.824743 + 0.565508i \(0.191320\pi\)
\(158\) 915.792 0.461117
\(159\) −1049.51 −0.523470
\(160\) 666.918 0.329528
\(161\) −3335.68 −1.63285
\(162\) −481.562 −0.233550
\(163\) 3281.47 1.57684 0.788418 0.615139i \(-0.210900\pi\)
0.788418 + 0.615139i \(0.210900\pi\)
\(164\) 891.818 0.424630
\(165\) −472.047 −0.222720
\(166\) −2114.66 −0.988731
\(167\) −3126.52 −1.44873 −0.724364 0.689418i \(-0.757866\pi\)
−0.724364 + 0.689418i \(0.757866\pi\)
\(168\) −4998.29 −2.29540
\(169\) 169.000 0.0769231
\(170\) 247.747 0.111773
\(171\) 1160.69 0.519066
\(172\) 841.474 0.373034
\(173\) 97.5698 0.0428792 0.0214396 0.999770i \(-0.493175\pi\)
0.0214396 + 0.999770i \(0.493175\pi\)
\(174\) 2982.31 1.29936
\(175\) 3052.40 1.31851
\(176\) 174.335 0.0746648
\(177\) −3818.06 −1.62137
\(178\) −686.900 −0.289243
\(179\) −34.7150 −0.0144956 −0.00724782 0.999974i \(-0.502307\pi\)
−0.00724782 + 0.999974i \(0.502307\pi\)
\(180\) 959.157 0.397174
\(181\) −1229.35 −0.504843 −0.252422 0.967617i \(-0.581227\pi\)
−0.252422 + 0.967617i \(0.581227\pi\)
\(182\) −551.700 −0.224696
\(183\) −1192.45 −0.481684
\(184\) 2599.25 1.04141
\(185\) −335.471 −0.133320
\(186\) −367.420 −0.144842
\(187\) 679.839 0.265854
\(188\) −2596.96 −1.00746
\(189\) −5056.40 −1.94603
\(190\) 133.309 0.0509012
\(191\) 4280.80 1.62172 0.810858 0.585243i \(-0.199001\pi\)
0.810858 + 0.585243i \(0.199001\pi\)
\(192\) 1745.81 0.656212
\(193\) 472.320 0.176157 0.0880786 0.996114i \(-0.471927\pi\)
0.0880786 + 0.996114i \(0.471927\pi\)
\(194\) 2359.93 0.873365
\(195\) 402.101 0.147667
\(196\) −2200.12 −0.801791
\(197\) −4484.37 −1.62182 −0.810908 0.585173i \(-0.801026\pi\)
−0.810908 + 0.585173i \(0.801026\pi\)
\(198\) −1154.00 −0.414197
\(199\) −366.240 −0.130463 −0.0652314 0.997870i \(-0.520779\pi\)
−0.0652314 + 0.997870i \(0.520779\pi\)
\(200\) −2378.51 −0.840931
\(201\) 4448.98 1.56123
\(202\) −525.087 −0.182896
\(203\) 5976.49 2.06634
\(204\) −2151.60 −0.738442
\(205\) 571.110 0.194576
\(206\) −503.932 −0.170440
\(207\) 5943.41 1.99563
\(208\) −148.503 −0.0495039
\(209\) 365.810 0.121070
\(210\) −1312.66 −0.431343
\(211\) 2122.55 0.692524 0.346262 0.938138i \(-0.387451\pi\)
0.346262 + 0.938138i \(0.387451\pi\)
\(212\) 672.095 0.217734
\(213\) 3566.95 1.14743
\(214\) −2240.23 −0.715603
\(215\) 538.870 0.170933
\(216\) 3940.08 1.24115
\(217\) −736.303 −0.230339
\(218\) −1325.99 −0.411960
\(219\) −2674.91 −0.825358
\(220\) 302.294 0.0926392
\(221\) −579.103 −0.176266
\(222\) −1277.39 −0.386185
\(223\) −5926.42 −1.77965 −0.889826 0.456301i \(-0.849174\pi\)
−0.889826 + 0.456301i \(0.849174\pi\)
\(224\) 5089.04 1.51797
\(225\) −5438.68 −1.61146
\(226\) −2521.17 −0.742060
\(227\) −895.661 −0.261881 −0.130941 0.991390i \(-0.541800\pi\)
−0.130941 + 0.991390i \(0.541800\pi\)
\(228\) −1157.74 −0.336286
\(229\) 627.717 0.181138 0.0905692 0.995890i \(-0.471131\pi\)
0.0905692 + 0.995890i \(0.471131\pi\)
\(230\) 682.617 0.195698
\(231\) −3602.04 −1.02596
\(232\) −4657.03 −1.31788
\(233\) 2303.72 0.647734 0.323867 0.946103i \(-0.395017\pi\)
0.323867 + 0.946103i \(0.395017\pi\)
\(234\) 983.002 0.274619
\(235\) −1663.06 −0.461643
\(236\) 2445.04 0.674401
\(237\) −5093.22 −1.39595
\(238\) 1890.48 0.514881
\(239\) 544.622 0.147400 0.0737001 0.997280i \(-0.476519\pi\)
0.0737001 + 0.997280i \(0.476519\pi\)
\(240\) −353.332 −0.0950313
\(241\) 5426.10 1.45031 0.725157 0.688584i \(-0.241767\pi\)
0.725157 + 0.688584i \(0.241767\pi\)
\(242\) 1714.73 0.455483
\(243\) −2345.23 −0.619121
\(244\) 763.629 0.200354
\(245\) −1408.93 −0.367400
\(246\) 2174.65 0.563621
\(247\) −311.606 −0.0802713
\(248\) 573.746 0.146907
\(249\) 11760.8 2.99321
\(250\) −1319.84 −0.333897
\(251\) −5221.22 −1.31299 −0.656494 0.754331i \(-0.727961\pi\)
−0.656494 + 0.754331i \(0.727961\pi\)
\(252\) 7319.02 1.82958
\(253\) 1873.16 0.465472
\(254\) −1351.02 −0.333741
\(255\) −1377.86 −0.338372
\(256\) −3457.26 −0.844057
\(257\) 658.206 0.159758 0.0798789 0.996805i \(-0.474547\pi\)
0.0798789 + 0.996805i \(0.474547\pi\)
\(258\) 2051.89 0.495136
\(259\) −2559.87 −0.614141
\(260\) −257.501 −0.0614213
\(261\) −10648.7 −2.52544
\(262\) 439.422 0.103617
\(263\) 3246.45 0.761160 0.380580 0.924748i \(-0.375724\pi\)
0.380580 + 0.924748i \(0.375724\pi\)
\(264\) 2806.81 0.654344
\(265\) 430.401 0.0997711
\(266\) 1017.24 0.234477
\(267\) 3820.23 0.875634
\(268\) −2849.07 −0.649384
\(269\) −2585.80 −0.586093 −0.293047 0.956098i \(-0.594669\pi\)
−0.293047 + 0.956098i \(0.594669\pi\)
\(270\) 1034.75 0.233233
\(271\) 988.933 0.221673 0.110836 0.993839i \(-0.464647\pi\)
0.110836 + 0.993839i \(0.464647\pi\)
\(272\) 508.867 0.113436
\(273\) 3068.31 0.680229
\(274\) 4124.74 0.909433
\(275\) −1714.09 −0.375866
\(276\) −5928.30 −1.29290
\(277\) 8142.40 1.76617 0.883086 0.469211i \(-0.155462\pi\)
0.883086 + 0.469211i \(0.155462\pi\)
\(278\) 3120.97 0.673322
\(279\) 1311.92 0.281515
\(280\) 2049.78 0.437493
\(281\) 1534.21 0.325705 0.162853 0.986650i \(-0.447930\pi\)
0.162853 + 0.986650i \(0.447930\pi\)
\(282\) −6332.54 −1.33722
\(283\) −6965.00 −1.46299 −0.731495 0.681847i \(-0.761177\pi\)
−0.731495 + 0.681847i \(0.761177\pi\)
\(284\) −2284.23 −0.477269
\(285\) −741.403 −0.154095
\(286\) 309.809 0.0640537
\(287\) 4357.96 0.896314
\(288\) −9067.49 −1.85523
\(289\) −2928.62 −0.596096
\(290\) −1223.04 −0.247652
\(291\) −13124.9 −2.64396
\(292\) 1712.98 0.343303
\(293\) 640.029 0.127614 0.0638070 0.997962i \(-0.479676\pi\)
0.0638070 + 0.997962i \(0.479676\pi\)
\(294\) −5364.87 −1.06424
\(295\) 1565.77 0.309027
\(296\) 1994.72 0.391691
\(297\) 2839.44 0.554750
\(298\) 2737.37 0.532120
\(299\) −1595.60 −0.308616
\(300\) 5424.85 1.04401
\(301\) 4111.95 0.787404
\(302\) 4363.99 0.831520
\(303\) 2920.30 0.553686
\(304\) 273.813 0.0516587
\(305\) 489.019 0.0918070
\(306\) −3368.40 −0.629276
\(307\) −100.406 −0.0186660 −0.00933299 0.999956i \(-0.502971\pi\)
−0.00933299 + 0.999956i \(0.502971\pi\)
\(308\) 2306.71 0.426743
\(309\) 2802.64 0.515977
\(310\) 150.678 0.0276062
\(311\) −3878.92 −0.707245 −0.353623 0.935388i \(-0.615050\pi\)
−0.353623 + 0.935388i \(0.615050\pi\)
\(312\) −2390.90 −0.433841
\(313\) −3789.39 −0.684311 −0.342155 0.939643i \(-0.611157\pi\)
−0.342155 + 0.939643i \(0.611157\pi\)
\(314\) −5067.04 −0.910668
\(315\) 4687.02 0.838360
\(316\) 3261.64 0.580638
\(317\) 4406.81 0.780791 0.390396 0.920647i \(-0.372338\pi\)
0.390396 + 0.920647i \(0.372338\pi\)
\(318\) 1638.87 0.289004
\(319\) −3356.11 −0.589048
\(320\) −715.950 −0.125071
\(321\) 12459.2 2.16636
\(322\) 5208.84 0.901482
\(323\) 1067.76 0.183938
\(324\) −1715.11 −0.294086
\(325\) 1460.10 0.249205
\(326\) −5124.19 −0.870559
\(327\) 7374.55 1.24714
\(328\) −3395.83 −0.571657
\(329\) −12690.3 −2.12656
\(330\) 737.127 0.122962
\(331\) −4131.49 −0.686064 −0.343032 0.939324i \(-0.611454\pi\)
−0.343032 + 0.939324i \(0.611454\pi\)
\(332\) −7531.47 −1.24501
\(333\) 4561.10 0.750591
\(334\) 4882.23 0.799831
\(335\) −1824.51 −0.297564
\(336\) −2696.17 −0.437762
\(337\) −4560.82 −0.737221 −0.368611 0.929584i \(-0.620166\pi\)
−0.368611 + 0.929584i \(0.620166\pi\)
\(338\) −263.902 −0.0424686
\(339\) 14021.6 2.24646
\(340\) 882.365 0.140744
\(341\) 413.473 0.0656622
\(342\) −1812.48 −0.286572
\(343\) −1429.34 −0.225007
\(344\) −3204.14 −0.502196
\(345\) −3796.41 −0.592441
\(346\) −152.360 −0.0236733
\(347\) 10069.4 1.55779 0.778896 0.627153i \(-0.215780\pi\)
0.778896 + 0.627153i \(0.215780\pi\)
\(348\) 10621.6 1.63615
\(349\) 5879.32 0.901757 0.450878 0.892585i \(-0.351111\pi\)
0.450878 + 0.892585i \(0.351111\pi\)
\(350\) −4766.49 −0.727942
\(351\) −2418.70 −0.367808
\(352\) −2857.76 −0.432725
\(353\) −9142.56 −1.37850 −0.689249 0.724525i \(-0.742059\pi\)
−0.689249 + 0.724525i \(0.742059\pi\)
\(354\) 5962.10 0.895147
\(355\) −1462.80 −0.218696
\(356\) −2446.43 −0.364215
\(357\) −10514.0 −1.55871
\(358\) 54.2093 0.00800293
\(359\) −2754.32 −0.404924 −0.202462 0.979290i \(-0.564894\pi\)
−0.202462 + 0.979290i \(0.564894\pi\)
\(360\) −3652.24 −0.534695
\(361\) −6284.45 −0.916235
\(362\) 1919.69 0.278720
\(363\) −9536.54 −1.37889
\(364\) −1964.91 −0.282937
\(365\) 1096.97 0.157310
\(366\) 1862.07 0.265934
\(367\) 3040.19 0.432416 0.216208 0.976347i \(-0.430631\pi\)
0.216208 + 0.976347i \(0.430631\pi\)
\(368\) 1402.08 0.198610
\(369\) −7764.88 −1.09546
\(370\) 523.855 0.0736052
\(371\) 3284.26 0.459596
\(372\) −1308.59 −0.182385
\(373\) −5384.72 −0.747481 −0.373740 0.927533i \(-0.621925\pi\)
−0.373740 + 0.927533i \(0.621925\pi\)
\(374\) −1061.60 −0.146776
\(375\) 7340.37 1.01081
\(376\) 9888.59 1.35629
\(377\) 2858.82 0.390548
\(378\) 7895.84 1.07439
\(379\) −3424.27 −0.464097 −0.232049 0.972704i \(-0.574543\pi\)
−0.232049 + 0.972704i \(0.574543\pi\)
\(380\) 474.786 0.0640948
\(381\) 7513.74 1.01034
\(382\) −6684.69 −0.895336
\(383\) −382.985 −0.0510956 −0.0255478 0.999674i \(-0.508133\pi\)
−0.0255478 + 0.999674i \(0.508133\pi\)
\(384\) 10283.8 1.36665
\(385\) 1477.19 0.195544
\(386\) −737.553 −0.0972551
\(387\) −7326.54 −0.962349
\(388\) 8405.00 1.09974
\(389\) 8588.34 1.11940 0.559699 0.828696i \(-0.310917\pi\)
0.559699 + 0.828696i \(0.310917\pi\)
\(390\) −627.902 −0.0815258
\(391\) 5467.56 0.707178
\(392\) 8377.52 1.07941
\(393\) −2443.87 −0.313681
\(394\) 7002.57 0.895392
\(395\) 2088.72 0.266063
\(396\) −4110.02 −0.521556
\(397\) −7239.16 −0.915171 −0.457586 0.889166i \(-0.651286\pi\)
−0.457586 + 0.889166i \(0.651286\pi\)
\(398\) 571.904 0.0720275
\(399\) −5657.41 −0.709837
\(400\) −1283.01 −0.160376
\(401\) 4269.62 0.531708 0.265854 0.964013i \(-0.414346\pi\)
0.265854 + 0.964013i \(0.414346\pi\)
\(402\) −6947.32 −0.861942
\(403\) −352.206 −0.0435351
\(404\) −1870.12 −0.230302
\(405\) −1098.33 −0.134757
\(406\) −9332.60 −1.14081
\(407\) 1437.50 0.175072
\(408\) 8192.78 0.994125
\(409\) 13562.5 1.63967 0.819834 0.572602i \(-0.194066\pi\)
0.819834 + 0.572602i \(0.194066\pi\)
\(410\) −891.818 −0.107424
\(411\) −22939.9 −2.75315
\(412\) −1794.78 −0.214618
\(413\) 11947.9 1.42353
\(414\) −9280.95 −1.10177
\(415\) −4823.06 −0.570494
\(416\) 2434.31 0.286904
\(417\) −17357.5 −2.03837
\(418\) −571.232 −0.0668418
\(419\) −14576.9 −1.69959 −0.849794 0.527114i \(-0.823274\pi\)
−0.849794 + 0.527114i \(0.823274\pi\)
\(420\) −4675.10 −0.543147
\(421\) 15848.4 1.83469 0.917343 0.398099i \(-0.130330\pi\)
0.917343 + 0.398099i \(0.130330\pi\)
\(422\) −3314.48 −0.382337
\(423\) 22611.2 2.59904
\(424\) −2559.18 −0.293124
\(425\) −5003.24 −0.571042
\(426\) −5569.98 −0.633490
\(427\) 3731.55 0.422909
\(428\) −7978.70 −0.901087
\(429\) −1723.02 −0.193912
\(430\) −841.474 −0.0943709
\(431\) 10694.7 1.19524 0.597618 0.801781i \(-0.296114\pi\)
0.597618 + 0.801781i \(0.296114\pi\)
\(432\) 2125.35 0.236703
\(433\) −16079.0 −1.78454 −0.892272 0.451498i \(-0.850890\pi\)
−0.892272 + 0.451498i \(0.850890\pi\)
\(434\) 1149.78 0.127168
\(435\) 6801.98 0.749724
\(436\) −4722.57 −0.518739
\(437\) 2942.01 0.322049
\(438\) 4177.01 0.455674
\(439\) 6035.80 0.656203 0.328101 0.944643i \(-0.393591\pi\)
0.328101 + 0.944643i \(0.393591\pi\)
\(440\) −1151.06 −0.124715
\(441\) 19156.0 2.06845
\(442\) 904.300 0.0973149
\(443\) 10201.3 1.09409 0.547043 0.837105i \(-0.315753\pi\)
0.547043 + 0.837105i \(0.315753\pi\)
\(444\) −4549.50 −0.486283
\(445\) −1566.66 −0.166892
\(446\) 9254.41 0.982532
\(447\) −15224.0 −1.61090
\(448\) −5463.19 −0.576141
\(449\) −5822.54 −0.611988 −0.305994 0.952033i \(-0.598989\pi\)
−0.305994 + 0.952033i \(0.598989\pi\)
\(450\) 8492.78 0.889675
\(451\) −2447.22 −0.255511
\(452\) −8979.27 −0.934402
\(453\) −24270.5 −2.51728
\(454\) 1398.62 0.144583
\(455\) −1258.30 −0.129649
\(456\) 4408.40 0.452724
\(457\) 4621.60 0.473062 0.236531 0.971624i \(-0.423990\pi\)
0.236531 + 0.971624i \(0.423990\pi\)
\(458\) −980.213 −0.100005
\(459\) 8288.03 0.842816
\(460\) 2431.18 0.246422
\(461\) 5127.77 0.518056 0.259028 0.965870i \(-0.416598\pi\)
0.259028 + 0.965870i \(0.416598\pi\)
\(462\) 5624.78 0.566425
\(463\) 6486.27 0.651064 0.325532 0.945531i \(-0.394457\pi\)
0.325532 + 0.945531i \(0.394457\pi\)
\(464\) −2512.09 −0.251338
\(465\) −838.004 −0.0835731
\(466\) −3597.39 −0.357609
\(467\) 12978.0 1.28598 0.642990 0.765875i \(-0.277694\pi\)
0.642990 + 0.765875i \(0.277694\pi\)
\(468\) 3501.01 0.345800
\(469\) −13922.3 −1.37073
\(470\) 2596.96 0.254869
\(471\) 28180.6 2.75689
\(472\) −9310.13 −0.907910
\(473\) −2309.08 −0.224464
\(474\) 7953.34 0.770694
\(475\) −2692.16 −0.260053
\(476\) 6733.04 0.648337
\(477\) −5851.79 −0.561709
\(478\) −850.456 −0.0813786
\(479\) −5808.96 −0.554109 −0.277055 0.960854i \(-0.589358\pi\)
−0.277055 + 0.960854i \(0.589358\pi\)
\(480\) 5791.95 0.550761
\(481\) −1224.50 −0.116076
\(482\) −8473.14 −0.800707
\(483\) −28969.2 −2.72908
\(484\) 6107.09 0.573543
\(485\) 5382.46 0.503928
\(486\) 3662.20 0.341812
\(487\) −5387.14 −0.501262 −0.250631 0.968083i \(-0.580638\pi\)
−0.250631 + 0.968083i \(0.580638\pi\)
\(488\) −2907.72 −0.269726
\(489\) 28498.4 2.63547
\(490\) 2200.12 0.202839
\(491\) 15259.1 1.40251 0.701255 0.712911i \(-0.252624\pi\)
0.701255 + 0.712911i \(0.252624\pi\)
\(492\) 7745.14 0.709711
\(493\) −9796.16 −0.894922
\(494\) 486.589 0.0443172
\(495\) −2632.01 −0.238990
\(496\) 309.489 0.0280171
\(497\) −11162.1 −1.00742
\(498\) −18365.1 −1.65253
\(499\) 1856.04 0.166509 0.0832544 0.996528i \(-0.473469\pi\)
0.0832544 + 0.996528i \(0.473469\pi\)
\(500\) −4700.69 −0.420442
\(501\) −27152.8 −2.42135
\(502\) 8153.20 0.724891
\(503\) 1049.46 0.0930283 0.0465142 0.998918i \(-0.485189\pi\)
0.0465142 + 0.998918i \(0.485189\pi\)
\(504\) −27869.1 −2.46307
\(505\) −1197.61 −0.105530
\(506\) −2925.04 −0.256984
\(507\) 1467.71 0.128566
\(508\) −4811.71 −0.420246
\(509\) −551.106 −0.0479909 −0.0239954 0.999712i \(-0.507639\pi\)
−0.0239954 + 0.999712i \(0.507639\pi\)
\(510\) 2151.60 0.186813
\(511\) 8370.64 0.724649
\(512\) −4074.36 −0.351686
\(513\) 4459.66 0.383818
\(514\) −1027.82 −0.0882010
\(515\) −1149.36 −0.0983431
\(516\) 7307.92 0.623475
\(517\) 7126.26 0.606214
\(518\) 3997.37 0.339063
\(519\) 847.361 0.0716667
\(520\) 980.503 0.0826883
\(521\) −8995.30 −0.756413 −0.378206 0.925721i \(-0.623459\pi\)
−0.378206 + 0.925721i \(0.623459\pi\)
\(522\) 16628.5 1.39427
\(523\) 2663.91 0.222724 0.111362 0.993780i \(-0.464479\pi\)
0.111362 + 0.993780i \(0.464479\pi\)
\(524\) 1565.02 0.130474
\(525\) 26509.1 2.20372
\(526\) −5069.51 −0.420230
\(527\) 1206.89 0.0997586
\(528\) 1514.04 0.124792
\(529\) 2897.77 0.238167
\(530\) −672.095 −0.0550829
\(531\) −21288.5 −1.73981
\(532\) 3622.94 0.295253
\(533\) 2084.60 0.169408
\(534\) −5965.49 −0.483431
\(535\) −5109.47 −0.412900
\(536\) 10848.6 0.874232
\(537\) −301.488 −0.0242275
\(538\) 4037.86 0.323577
\(539\) 6037.30 0.482458
\(540\) 3685.31 0.293686
\(541\) −6169.23 −0.490270 −0.245135 0.969489i \(-0.578832\pi\)
−0.245135 + 0.969489i \(0.578832\pi\)
\(542\) −1544.27 −0.122384
\(543\) −10676.5 −0.843776
\(544\) −8341.52 −0.657426
\(545\) −3024.28 −0.237699
\(546\) −4791.32 −0.375549
\(547\) 5140.42 0.401807 0.200904 0.979611i \(-0.435612\pi\)
0.200904 + 0.979611i \(0.435612\pi\)
\(548\) 14690.5 1.14516
\(549\) −6648.76 −0.516871
\(550\) 2676.64 0.207513
\(551\) −5271.15 −0.407547
\(552\) 22573.6 1.74057
\(553\) 15938.3 1.22562
\(554\) −12714.8 −0.975090
\(555\) −2913.45 −0.222827
\(556\) 11115.5 0.847847
\(557\) 2778.56 0.211367 0.105683 0.994400i \(-0.466297\pi\)
0.105683 + 0.994400i \(0.466297\pi\)
\(558\) −2048.64 −0.155422
\(559\) 1966.93 0.148823
\(560\) 1105.69 0.0834356
\(561\) 5904.17 0.444339
\(562\) −2395.75 −0.179819
\(563\) 4906.14 0.367263 0.183632 0.982995i \(-0.441215\pi\)
0.183632 + 0.982995i \(0.441215\pi\)
\(564\) −22553.7 −1.68383
\(565\) −5750.22 −0.428166
\(566\) 10876.2 0.807706
\(567\) −8381.04 −0.620759
\(568\) 8697.82 0.642522
\(569\) −9363.15 −0.689849 −0.344924 0.938631i \(-0.612095\pi\)
−0.344924 + 0.938631i \(0.612095\pi\)
\(570\) 1157.74 0.0850744
\(571\) 7199.32 0.527640 0.263820 0.964572i \(-0.415018\pi\)
0.263820 + 0.964572i \(0.415018\pi\)
\(572\) 1103.40 0.0806564
\(573\) 37177.3 2.71048
\(574\) −6805.18 −0.494848
\(575\) −13785.4 −0.999813
\(576\) 9734.14 0.704148
\(577\) −11449.6 −0.826086 −0.413043 0.910711i \(-0.635534\pi\)
−0.413043 + 0.910711i \(0.635534\pi\)
\(578\) 4573.19 0.329100
\(579\) 4101.94 0.294423
\(580\) −4355.91 −0.311843
\(581\) −36803.3 −2.62798
\(582\) 20495.2 1.45971
\(583\) −1844.28 −0.131016
\(584\) −6522.62 −0.462171
\(585\) 2242.01 0.158454
\(586\) −999.439 −0.0704547
\(587\) −5439.39 −0.382466 −0.191233 0.981545i \(-0.561249\pi\)
−0.191233 + 0.981545i \(0.561249\pi\)
\(588\) −19107.3 −1.34008
\(589\) 649.406 0.0454301
\(590\) −2445.04 −0.170611
\(591\) −38945.2 −2.71064
\(592\) 1075.99 0.0747006
\(593\) −28405.8 −1.96709 −0.983547 0.180651i \(-0.942180\pi\)
−0.983547 + 0.180651i \(0.942180\pi\)
\(594\) −4433.93 −0.306273
\(595\) 4311.76 0.297084
\(596\) 9749.30 0.670045
\(597\) −3180.67 −0.218051
\(598\) 2491.62 0.170384
\(599\) −10482.3 −0.715020 −0.357510 0.933909i \(-0.616374\pi\)
−0.357510 + 0.933909i \(0.616374\pi\)
\(600\) −20656.6 −1.40550
\(601\) 3199.54 0.217158 0.108579 0.994088i \(-0.465370\pi\)
0.108579 + 0.994088i \(0.465370\pi\)
\(602\) −6421.02 −0.434720
\(603\) 24806.3 1.67527
\(604\) 15542.6 1.04705
\(605\) 3910.91 0.262812
\(606\) −4560.20 −0.305686
\(607\) 11342.8 0.758468 0.379234 0.925301i \(-0.376188\pi\)
0.379234 + 0.925301i \(0.376188\pi\)
\(608\) −4488.44 −0.299392
\(609\) 51903.7 3.45361
\(610\) −763.629 −0.0506859
\(611\) −6070.32 −0.401930
\(612\) −11996.7 −0.792384
\(613\) 14385.4 0.947831 0.473916 0.880570i \(-0.342840\pi\)
0.473916 + 0.880570i \(0.342840\pi\)
\(614\) 156.789 0.0103053
\(615\) 4959.89 0.325207
\(616\) −8783.39 −0.574502
\(617\) 22056.8 1.43918 0.719588 0.694401i \(-0.244331\pi\)
0.719588 + 0.694401i \(0.244331\pi\)
\(618\) −4376.48 −0.284867
\(619\) 13621.4 0.884477 0.442238 0.896898i \(-0.354185\pi\)
0.442238 + 0.896898i \(0.354185\pi\)
\(620\) 536.648 0.0347618
\(621\) 22836.0 1.47565
\(622\) 6057.14 0.390465
\(623\) −11954.7 −0.768789
\(624\) −1289.70 −0.0827390
\(625\) 11029.2 0.705866
\(626\) 5917.34 0.377803
\(627\) 3176.94 0.202352
\(628\) −18046.5 −1.14671
\(629\) 4195.92 0.265982
\(630\) −7319.02 −0.462852
\(631\) −18737.5 −1.18214 −0.591068 0.806622i \(-0.701293\pi\)
−0.591068 + 0.806622i \(0.701293\pi\)
\(632\) −12419.6 −0.781683
\(633\) 18433.7 1.15746
\(634\) −6881.46 −0.431069
\(635\) −3081.36 −0.192567
\(636\) 5836.91 0.363913
\(637\) −5142.72 −0.319877
\(638\) 5240.75 0.325209
\(639\) 19888.4 1.23125
\(640\) −4217.35 −0.260477
\(641\) 29798.7 1.83616 0.918081 0.396394i \(-0.129739\pi\)
0.918081 + 0.396394i \(0.129739\pi\)
\(642\) −19455.6 −1.19603
\(643\) −22983.5 −1.40961 −0.704807 0.709399i \(-0.748966\pi\)
−0.704807 + 0.709399i \(0.748966\pi\)
\(644\) 18551.5 1.13515
\(645\) 4679.90 0.285692
\(646\) −1667.37 −0.101551
\(647\) −24905.4 −1.51334 −0.756672 0.653794i \(-0.773176\pi\)
−0.756672 + 0.653794i \(0.773176\pi\)
\(648\) 6530.72 0.395912
\(649\) −6709.39 −0.405804
\(650\) −2280.02 −0.137584
\(651\) −6394.54 −0.384980
\(652\) −18250.1 −1.09621
\(653\) 10077.8 0.603946 0.301973 0.953316i \(-0.402355\pi\)
0.301973 + 0.953316i \(0.402355\pi\)
\(654\) −11515.7 −0.688534
\(655\) 1002.22 0.0597864
\(656\) −1831.77 −0.109022
\(657\) −14914.6 −0.885650
\(658\) 19816.5 1.17406
\(659\) 12334.6 0.729116 0.364558 0.931181i \(-0.381220\pi\)
0.364558 + 0.931181i \(0.381220\pi\)
\(660\) 2625.32 0.154834
\(661\) −12749.1 −0.750202 −0.375101 0.926984i \(-0.622392\pi\)
−0.375101 + 0.926984i \(0.622392\pi\)
\(662\) 6451.54 0.378771
\(663\) −5029.31 −0.294604
\(664\) 28678.1 1.67609
\(665\) 2320.09 0.135292
\(666\) −7122.40 −0.414395
\(667\) −26991.3 −1.56688
\(668\) 17388.3 1.00715
\(669\) −51468.9 −2.97444
\(670\) 2849.07 0.164283
\(671\) −2095.46 −0.120558
\(672\) 44196.5 2.53708
\(673\) −13618.2 −0.780007 −0.390004 0.920813i \(-0.627526\pi\)
−0.390004 + 0.920813i \(0.627526\pi\)
\(674\) 7121.96 0.407014
\(675\) −20896.7 −1.19158
\(676\) −939.902 −0.0534765
\(677\) 9655.67 0.548150 0.274075 0.961708i \(-0.411628\pi\)
0.274075 + 0.961708i \(0.411628\pi\)
\(678\) −21895.5 −1.24025
\(679\) 41071.9 2.32135
\(680\) −3359.84 −0.189476
\(681\) −7778.51 −0.437699
\(682\) −645.660 −0.0362516
\(683\) 16316.8 0.914119 0.457060 0.889436i \(-0.348903\pi\)
0.457060 + 0.889436i \(0.348903\pi\)
\(684\) −6455.25 −0.360852
\(685\) 9407.61 0.524739
\(686\) 2232.00 0.124225
\(687\) 5451.51 0.302748
\(688\) −1728.37 −0.0957753
\(689\) 1571.01 0.0868658
\(690\) 5928.30 0.327082
\(691\) 2350.84 0.129421 0.0647106 0.997904i \(-0.479388\pi\)
0.0647106 + 0.997904i \(0.479388\pi\)
\(692\) −542.640 −0.0298093
\(693\) −20084.0 −1.10091
\(694\) −15723.9 −0.860045
\(695\) 7118.24 0.388504
\(696\) −40444.7 −2.20266
\(697\) −7143.20 −0.388189
\(698\) −9180.88 −0.497853
\(699\) 20007.1 1.08260
\(700\) −16976.1 −0.916623
\(701\) −8076.90 −0.435179 −0.217589 0.976040i \(-0.569819\pi\)
−0.217589 + 0.976040i \(0.569819\pi\)
\(702\) 3776.93 0.203064
\(703\) 2257.76 0.121128
\(704\) 3067.87 0.164239
\(705\) −14443.1 −0.771573
\(706\) 14276.6 0.761058
\(707\) −9138.55 −0.486125
\(708\) 21234.3 1.12717
\(709\) −13624.9 −0.721712 −0.360856 0.932622i \(-0.617515\pi\)
−0.360856 + 0.932622i \(0.617515\pi\)
\(710\) 2284.23 0.120741
\(711\) −28398.4 −1.49792
\(712\) 9315.43 0.490324
\(713\) 3325.33 0.174663
\(714\) 16418.2 0.860553
\(715\) 706.604 0.0369587
\(716\) 193.069 0.0100773
\(717\) 4729.86 0.246359
\(718\) 4301.02 0.223555
\(719\) 16235.8 0.842131 0.421066 0.907030i \(-0.361656\pi\)
0.421066 + 0.907030i \(0.361656\pi\)
\(720\) −1970.09 −0.101973
\(721\) −8770.37 −0.453018
\(722\) 9813.51 0.505846
\(723\) 47123.8 2.42400
\(724\) 6837.08 0.350964
\(725\) 24699.2 1.26525
\(726\) 14891.8 0.761277
\(727\) 24181.2 1.23361 0.616803 0.787118i \(-0.288428\pi\)
0.616803 + 0.787118i \(0.288428\pi\)
\(728\) 7481.91 0.380904
\(729\) −28693.9 −1.45780
\(730\) −1712.98 −0.0868496
\(731\) −6739.96 −0.341021
\(732\) 6631.86 0.334864
\(733\) 3053.70 0.153876 0.0769379 0.997036i \(-0.475486\pi\)
0.0769379 + 0.997036i \(0.475486\pi\)
\(734\) −4747.41 −0.238733
\(735\) −12236.1 −0.614060
\(736\) −22983.4 −1.15106
\(737\) 7818.10 0.390751
\(738\) 12125.3 0.604793
\(739\) −8033.62 −0.399894 −0.199947 0.979807i \(-0.564077\pi\)
−0.199947 + 0.979807i \(0.564077\pi\)
\(740\) 1865.74 0.0926836
\(741\) −2706.19 −0.134163
\(742\) −5128.54 −0.253739
\(743\) −16139.6 −0.796912 −0.398456 0.917187i \(-0.630454\pi\)
−0.398456 + 0.917187i \(0.630454\pi\)
\(744\) 4982.79 0.245535
\(745\) 6243.33 0.307031
\(746\) 8408.53 0.412678
\(747\) 65574.9 3.21186
\(748\) −3780.96 −0.184820
\(749\) −38988.7 −1.90202
\(750\) −11462.4 −0.558062
\(751\) −18491.1 −0.898469 −0.449235 0.893414i \(-0.648303\pi\)
−0.449235 + 0.893414i \(0.648303\pi\)
\(752\) 5334.08 0.258662
\(753\) −45344.5 −2.19448
\(754\) −4464.20 −0.215619
\(755\) 9953.28 0.479784
\(756\) 28121.5 1.35287
\(757\) 160.630 0.00771227 0.00385613 0.999993i \(-0.498773\pi\)
0.00385613 + 0.999993i \(0.498773\pi\)
\(758\) 5347.17 0.256224
\(759\) 16267.7 0.777973
\(760\) −1807.87 −0.0862874
\(761\) 26799.1 1.27656 0.638282 0.769803i \(-0.279645\pi\)
0.638282 + 0.769803i \(0.279645\pi\)
\(762\) −11733.1 −0.557803
\(763\) −23077.3 −1.09496
\(764\) −23807.9 −1.12741
\(765\) −7682.57 −0.363090
\(766\) 598.052 0.0282095
\(767\) 5715.22 0.269054
\(768\) −30025.1 −1.41073
\(769\) −5145.82 −0.241304 −0.120652 0.992695i \(-0.538499\pi\)
−0.120652 + 0.992695i \(0.538499\pi\)
\(770\) −2306.71 −0.107958
\(771\) 5716.29 0.267013
\(772\) −2626.83 −0.122463
\(773\) 12810.6 0.596072 0.298036 0.954555i \(-0.403668\pi\)
0.298036 + 0.954555i \(0.403668\pi\)
\(774\) 11440.8 0.531306
\(775\) −3042.94 −0.141039
\(776\) −32004.3 −1.48052
\(777\) −22231.6 −1.02645
\(778\) −13411.1 −0.618011
\(779\) −3843.64 −0.176781
\(780\) −2236.31 −0.102657
\(781\) 6268.13 0.287185
\(782\) −8537.89 −0.390428
\(783\) −40915.0 −1.86741
\(784\) 4518.98 0.205857
\(785\) −11556.8 −0.525452
\(786\) 3816.23 0.173181
\(787\) 28073.0 1.27153 0.635764 0.771883i \(-0.280685\pi\)
0.635764 + 0.771883i \(0.280685\pi\)
\(788\) 24940.0 1.12748
\(789\) 28194.3 1.27217
\(790\) −3261.64 −0.146891
\(791\) −43878.1 −1.97235
\(792\) 15650.0 0.702144
\(793\) 1784.96 0.0799318
\(794\) 11304.3 0.505259
\(795\) 3737.89 0.166754
\(796\) 2036.87 0.0906970
\(797\) 30093.1 1.33746 0.668729 0.743507i \(-0.266839\pi\)
0.668729 + 0.743507i \(0.266839\pi\)
\(798\) 8834.35 0.391896
\(799\) 20800.8 0.921003
\(800\) 21031.6 0.929473
\(801\) 21300.6 0.939598
\(802\) −6667.24 −0.293552
\(803\) −4700.56 −0.206574
\(804\) −24743.2 −1.08536
\(805\) 11880.2 0.520151
\(806\) 549.989 0.0240354
\(807\) −22456.8 −0.979575
\(808\) 7120.99 0.310044
\(809\) −24337.1 −1.05766 −0.528831 0.848727i \(-0.677369\pi\)
−0.528831 + 0.848727i \(0.677369\pi\)
\(810\) 1715.11 0.0743984
\(811\) 19078.7 0.826071 0.413035 0.910715i \(-0.364469\pi\)
0.413035 + 0.910715i \(0.364469\pi\)
\(812\) −33238.5 −1.43651
\(813\) 8588.54 0.370496
\(814\) −2244.74 −0.0966559
\(815\) −11687.1 −0.502309
\(816\) 4419.33 0.189593
\(817\) −3626.66 −0.155301
\(818\) −21178.6 −0.905248
\(819\) 17108.0 0.729919
\(820\) −3176.26 −0.135268
\(821\) 2013.92 0.0856104 0.0428052 0.999083i \(-0.486371\pi\)
0.0428052 + 0.999083i \(0.486371\pi\)
\(822\) 35821.9 1.51999
\(823\) −7692.10 −0.325795 −0.162898 0.986643i \(-0.552084\pi\)
−0.162898 + 0.986643i \(0.552084\pi\)
\(824\) 6834.10 0.288929
\(825\) −14886.2 −0.628209
\(826\) −18657.3 −0.785922
\(827\) −4762.76 −0.200263 −0.100131 0.994974i \(-0.531926\pi\)
−0.100131 + 0.994974i \(0.531926\pi\)
\(828\) −33054.6 −1.38735
\(829\) 19977.7 0.836976 0.418488 0.908222i \(-0.362560\pi\)
0.418488 + 0.908222i \(0.362560\pi\)
\(830\) 7531.47 0.314965
\(831\) 70714.0 2.95191
\(832\) −2613.28 −0.108893
\(833\) 17622.3 0.732984
\(834\) 27104.6 1.12537
\(835\) 11135.3 0.461499
\(836\) −2034.47 −0.0841671
\(837\) 5040.72 0.208164
\(838\) 22762.6 0.938330
\(839\) −30615.8 −1.25980 −0.629901 0.776676i \(-0.716905\pi\)
−0.629901 + 0.776676i \(0.716905\pi\)
\(840\) 17801.7 0.731210
\(841\) 23971.0 0.982861
\(842\) −24748.1 −1.01292
\(843\) 13324.1 0.544372
\(844\) −11804.7 −0.481439
\(845\) −601.902 −0.0245042
\(846\) −35308.5 −1.43491
\(847\) 29842.9 1.21064
\(848\) −1380.47 −0.0559026
\(849\) −60488.6 −2.44519
\(850\) 7812.83 0.315268
\(851\) 11561.0 0.465696
\(852\) −19837.8 −0.797690
\(853\) 5660.88 0.227227 0.113614 0.993525i \(-0.463757\pi\)
0.113614 + 0.993525i \(0.463757\pi\)
\(854\) −5827.01 −0.233485
\(855\) −4133.86 −0.165351
\(856\) 30381.0 1.21309
\(857\) 41346.1 1.64802 0.824012 0.566572i \(-0.191731\pi\)
0.824012 + 0.566572i \(0.191731\pi\)
\(858\) 2690.58 0.107057
\(859\) −34810.5 −1.38268 −0.691339 0.722530i \(-0.742979\pi\)
−0.691339 + 0.722530i \(0.742979\pi\)
\(860\) −2996.96 −0.118832
\(861\) 37847.4 1.49807
\(862\) −16700.4 −0.659880
\(863\) −8360.51 −0.329774 −0.164887 0.986312i \(-0.552726\pi\)
−0.164887 + 0.986312i \(0.552726\pi\)
\(864\) −34839.5 −1.37183
\(865\) −347.500 −0.0136594
\(866\) 25108.2 0.985233
\(867\) −25434.1 −0.996293
\(868\) 4094.99 0.160130
\(869\) −8950.21 −0.349385
\(870\) −10621.6 −0.413917
\(871\) −6659.64 −0.259074
\(872\) 17982.4 0.698352
\(873\) −73180.6 −2.83710
\(874\) −4594.10 −0.177801
\(875\) −22970.4 −0.887475
\(876\) 14876.6 0.573784
\(877\) −40579.3 −1.56245 −0.781223 0.624251i \(-0.785404\pi\)
−0.781223 + 0.624251i \(0.785404\pi\)
\(878\) −9425.22 −0.362285
\(879\) 5558.43 0.213289
\(880\) −620.903 −0.0237848
\(881\) 10445.2 0.399442 0.199721 0.979853i \(-0.435996\pi\)
0.199721 + 0.979853i \(0.435996\pi\)
\(882\) −29913.0 −1.14198
\(883\) 18227.6 0.694685 0.347343 0.937738i \(-0.387084\pi\)
0.347343 + 0.937738i \(0.387084\pi\)
\(884\) 3220.71 0.122539
\(885\) 13598.2 0.516496
\(886\) −15929.9 −0.604036
\(887\) −23517.7 −0.890245 −0.445122 0.895470i \(-0.646840\pi\)
−0.445122 + 0.895470i \(0.646840\pi\)
\(888\) 17323.4 0.654658
\(889\) −23512.9 −0.887061
\(890\) 2446.43 0.0921399
\(891\) 4706.40 0.176959
\(892\) 32960.1 1.23720
\(893\) 11192.6 0.419424
\(894\) 23773.1 0.889366
\(895\) 123.639 0.00461766
\(896\) −32181.2 −1.19989
\(897\) −13857.3 −0.515809
\(898\) 9092.21 0.337874
\(899\) −5957.95 −0.221033
\(900\) 30247.5 1.12028
\(901\) −5383.28 −0.199049
\(902\) 3821.47 0.141065
\(903\) 35710.9 1.31604
\(904\) 34191.0 1.25794
\(905\) 4378.38 0.160820
\(906\) 37899.7 1.38977
\(907\) −30564.6 −1.11894 −0.559471 0.828850i \(-0.688996\pi\)
−0.559471 + 0.828850i \(0.688996\pi\)
\(908\) 4981.26 0.182058
\(909\) 16282.8 0.594132
\(910\) 1964.91 0.0715781
\(911\) −32766.5 −1.19166 −0.595831 0.803110i \(-0.703177\pi\)
−0.595831 + 0.803110i \(0.703177\pi\)
\(912\) 2377.97 0.0863404
\(913\) 20667.0 0.749154
\(914\) −7216.87 −0.261174
\(915\) 4246.96 0.153443
\(916\) −3491.08 −0.125926
\(917\) 7647.64 0.275406
\(918\) −12942.2 −0.465312
\(919\) 20686.7 0.742538 0.371269 0.928525i \(-0.378923\pi\)
0.371269 + 0.928525i \(0.378923\pi\)
\(920\) −9257.35 −0.331745
\(921\) −871.989 −0.0311976
\(922\) −8007.28 −0.286015
\(923\) −5339.34 −0.190408
\(924\) 20033.0 0.713242
\(925\) −10579.2 −0.376047
\(926\) −10128.6 −0.359447
\(927\) 15626.8 0.553669
\(928\) 41179.0 1.45665
\(929\) 45632.2 1.61156 0.805782 0.592212i \(-0.201745\pi\)
0.805782 + 0.592212i \(0.201745\pi\)
\(930\) 1308.59 0.0461401
\(931\) 9482.26 0.333801
\(932\) −12812.3 −0.450301
\(933\) −33687.1 −1.18206
\(934\) −20265.9 −0.709979
\(935\) −2421.28 −0.0846892
\(936\) −13331.0 −0.465532
\(937\) −17761.4 −0.619253 −0.309626 0.950858i \(-0.600204\pi\)
−0.309626 + 0.950858i \(0.600204\pi\)
\(938\) 21740.4 0.756768
\(939\) −32909.6 −1.14373
\(940\) 9249.19 0.320931
\(941\) −44888.3 −1.55507 −0.777534 0.628841i \(-0.783530\pi\)
−0.777534 + 0.628841i \(0.783530\pi\)
\(942\) −44005.5 −1.52206
\(943\) −19681.7 −0.679664
\(944\) −5022.05 −0.173150
\(945\) 18008.6 0.619917
\(946\) 3605.74 0.123925
\(947\) 16069.6 0.551415 0.275708 0.961242i \(-0.411088\pi\)
0.275708 + 0.961242i \(0.411088\pi\)
\(948\) 28326.2 0.970457
\(949\) 4004.05 0.136962
\(950\) 4203.96 0.143573
\(951\) 38271.6 1.30499
\(952\) −25637.8 −0.872823
\(953\) 3512.03 0.119377 0.0596883 0.998217i \(-0.480989\pi\)
0.0596883 + 0.998217i \(0.480989\pi\)
\(954\) 9137.88 0.310115
\(955\) −15246.3 −0.516605
\(956\) −3028.94 −0.102472
\(957\) −29146.7 −0.984513
\(958\) 9071.00 0.305919
\(959\) 71786.5 2.41721
\(960\) −6217.78 −0.209040
\(961\) −29057.0 −0.975361
\(962\) 1912.12 0.0640844
\(963\) 69468.9 2.32461
\(964\) −30177.5 −1.00825
\(965\) −1682.19 −0.0561158
\(966\) 45237.0 1.50670
\(967\) −37011.9 −1.23084 −0.615421 0.788199i \(-0.711014\pi\)
−0.615421 + 0.788199i \(0.711014\pi\)
\(968\) −23254.4 −0.772132
\(969\) 9273.16 0.307427
\(970\) −8405.00 −0.278215
\(971\) 19532.3 0.645542 0.322771 0.946477i \(-0.395386\pi\)
0.322771 + 0.946477i \(0.395386\pi\)
\(972\) 13043.1 0.430409
\(973\) 54317.1 1.78965
\(974\) 8412.30 0.276743
\(975\) 12680.5 0.416513
\(976\) −1568.47 −0.0514402
\(977\) 30201.2 0.988970 0.494485 0.869186i \(-0.335357\pi\)
0.494485 + 0.869186i \(0.335357\pi\)
\(978\) −44501.8 −1.45502
\(979\) 6713.21 0.219157
\(980\) 7835.83 0.255415
\(981\) 41118.5 1.33824
\(982\) −23827.8 −0.774315
\(983\) −38774.9 −1.25812 −0.629058 0.777359i \(-0.716559\pi\)
−0.629058 + 0.777359i \(0.716559\pi\)
\(984\) −29491.7 −0.955447
\(985\) 15971.3 0.516638
\(986\) 15297.2 0.494080
\(987\) −110211. −3.55425
\(988\) 1733.01 0.0558041
\(989\) −18570.6 −0.597079
\(990\) 4110.02 0.131944
\(991\) −27728.9 −0.888838 −0.444419 0.895819i \(-0.646590\pi\)
−0.444419 + 0.895819i \(0.646590\pi\)
\(992\) −5073.25 −0.162375
\(993\) −35880.6 −1.14666
\(994\) 17430.3 0.556192
\(995\) 1304.38 0.0415596
\(996\) −65408.2 −2.08086
\(997\) −48918.2 −1.55392 −0.776958 0.629552i \(-0.783239\pi\)
−0.776958 + 0.629552i \(0.783239\pi\)
\(998\) −2898.31 −0.0919283
\(999\) 17524.8 0.555016
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 13.4.a.b.1.1 2
3.2 odd 2 117.4.a.d.1.2 2
4.3 odd 2 208.4.a.h.1.1 2
5.2 odd 4 325.4.b.e.274.2 4
5.3 odd 4 325.4.b.e.274.3 4
5.4 even 2 325.4.a.f.1.2 2
7.6 odd 2 637.4.a.b.1.1 2
8.3 odd 2 832.4.a.z.1.2 2
8.5 even 2 832.4.a.s.1.1 2
11.10 odd 2 1573.4.a.b.1.2 2
12.11 even 2 1872.4.a.bb.1.2 2
13.2 odd 12 169.4.e.f.147.2 8
13.3 even 3 169.4.c.g.22.2 4
13.4 even 6 169.4.c.j.146.1 4
13.5 odd 4 169.4.b.f.168.3 4
13.6 odd 12 169.4.e.f.23.3 8
13.7 odd 12 169.4.e.f.23.2 8
13.8 odd 4 169.4.b.f.168.2 4
13.9 even 3 169.4.c.g.146.2 4
13.10 even 6 169.4.c.j.22.1 4
13.11 odd 12 169.4.e.f.147.3 8
13.12 even 2 169.4.a.g.1.2 2
39.38 odd 2 1521.4.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.b.1.1 2 1.1 even 1 trivial
117.4.a.d.1.2 2 3.2 odd 2
169.4.a.g.1.2 2 13.12 even 2
169.4.b.f.168.2 4 13.8 odd 4
169.4.b.f.168.3 4 13.5 odd 4
169.4.c.g.22.2 4 13.3 even 3
169.4.c.g.146.2 4 13.9 even 3
169.4.c.j.22.1 4 13.10 even 6
169.4.c.j.146.1 4 13.4 even 6
169.4.e.f.23.2 8 13.7 odd 12
169.4.e.f.23.3 8 13.6 odd 12
169.4.e.f.147.2 8 13.2 odd 12
169.4.e.f.147.3 8 13.11 odd 12
208.4.a.h.1.1 2 4.3 odd 2
325.4.a.f.1.2 2 5.4 even 2
325.4.b.e.274.2 4 5.2 odd 4
325.4.b.e.274.3 4 5.3 odd 4
637.4.a.b.1.1 2 7.6 odd 2
832.4.a.s.1.1 2 8.5 even 2
832.4.a.z.1.2 2 8.3 odd 2
1521.4.a.r.1.1 2 39.38 odd 2
1573.4.a.b.1.2 2 11.10 odd 2
1872.4.a.bb.1.2 2 12.11 even 2